A martingale is a mathematical equivalent of a fair game. A simple example is a coin toss whereby the coin has equal chance of landing head or tail. As would normally be expected, the coin is not biased (as it might be if coin sized discs of materials of different densities are welded together). After a large number of coin-tosses we would end up with roughly the same number of head results as tail results.

Martingales with respect to filtration

[ref]Bernd Schmid, Credit Risk Pricing Models Theory and Practice. Second edition[/ref]

A stochastic process $M$ on a filtered probability space is called a martingale with respect to a filtration and with respect to $P$ if

• $M_t$ is $\mathcal{F}_t$-measurable for all $t\leq0$
• $E^P[|M_t|] < \infty$ for all $t\leq0$